A mixed interpolation-regression method for numerical integration on the unit circle using zeros of para-orthogonal polynomials

TL;DR

This paper introduces a novel numerical integration method on the unit circle that combines interpolation and regression techniques using zeros of para-orthogonal polynomials, improving accuracy over traditional Szeg\

Contribution

It proposes a mixed interpolation-regression approach utilizing zeros of para-orthogonal polynomials for more accurate integral estimation on the unit circle.

Findings

The method effectively approximates integrals with fewer points.

Numerical examples demonstrate improved accuracy.

The approach is applicable with uniformly distributed data points.

Abstract

A new alternative numerical procedure to the Szeg\H{o} quadrature formulas for the estimation of integrals with respect to a positive Borel measure $\mu$ supported on the unit circle is presented. As in many practical situations, we assume that the values of the integrand $F$ are only known at a finite number of points, which we will assume to be uniformly distributed on the unit circle (although this does not actually constitute a restriction). Our technique consists of obtaining an approximating Laurent polynomial $L$ to $F$ by interpolation in the Hermite sense in a collection of these points that mimic the zeros of a para-orthogonal polynomial with respect to $\mu$, and to use the values of $F$ at the remaining nodes to improve the accuracy of the approximation by a process of simultaneous complex regression. Some numerical examples are carried out.